218 lines
8.4 KiB
Python
218 lines
8.4 KiB
Python
class as_polyform:
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def __init__(self, form, mult):
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'''Elements of H^0(Ω^⊗n). Usage: mult is n, form should be as_function.'''
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self.form = form
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self.curve = form.curve
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self.mult = mult
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def __add__(self, other):
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return as_polyform(self.form + other.form, self.mult)
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def __repr__(self):
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return '(' + str(self.form) + ') dx⊗' + str(self.mult)
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def expansion_at_infty(self):
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return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
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def coordinates(self, basis = 0):
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"""Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
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AS = self.curve
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if basis == 0:
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basis = AS.holo_polydifferentials_basis(mult = self.mult)
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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# We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
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# and sometimes basis elements have denominators. Thus we multiply by them.
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denom = LCM([denominator(omega.form.function) for omega in basis])
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basis = [denom*omega.form.function for omega in basis]
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self_with_no_denominator = denom*self.form.function
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return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
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def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
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'''Give the basis of H^0(Ω^⊗n) for n = mult.'''
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v = AS.dx.valuation()
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result = AS.at_most_poles(mult*v, threshold=threshold)
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result = [as_polyform(omega, mult) for omega in result]
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if mult == 1 and len(result) < AS.genus():
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raise ValueError('Increase threshold, not all forms found.')
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if mult > 1 and len(result) < (2*mult - 1)*(AS.genus() - 1):
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raise ValueError('Increase threshold, not all forms found.')
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return result
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as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis
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def as_symmetric_power_basis(AS, n, threshold = 8):
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'''Give the basis of H^0(Ω)^⊙n for n = mult.'''
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g = AS.genus()
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B0 = AS.holomorphic_differentials_basis(threshold=threshold)
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from itertools import product
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indices = [list(range(g)) for i in range(n)]
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indices_nonrepeating = []
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for i in product(*indices):
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if non_decreasing(i):
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indices_nonrepeating += [i]
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result = []
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for i in indices_nonrepeating:
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tensor_form = [B0[i[j]] for j in range(n)]
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tensor_form = [1] + tensor_form
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result += [tensor_form]
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result = [as_symmetric_product_forms([a]) for a in result]
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return result
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def as_canonical_ideal(AS, n, threshold=8):
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'''Return the n-th homogeneous part of the canonical ideal.'''
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B0 = AS.holomorphic_differentials_basis(threshold=threshold)
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F = AS.base_ring
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g = AS.genus()
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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from itertools import product
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B1 = as_symmetric_power_basis(AS, n, threshold = 8)
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B2 = AS.holo_polydifferentials_basis(n, threshold = threshold)
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g = AS.genus()
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r = len(B2)
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M = matrix(F, len(B1), r)
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for i in range(0, len(B1)):
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c = B1[i].multiply()
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#return M, c.coordinates(basis=B2)
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M[i, :] = vector(c.coordinates(basis=B2))
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K = M.kernel().basis()
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result = []
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for v in K:
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kernel_vector = 0*B1[0]
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for i in range(len(v)):
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kernel_vector += v[i]*B1[i]
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result += [kernel_vector]
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return result
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as_cover.canonical_ideal = as_canonical_ideal
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def as_canonical_ideal_polynomials(AS, n, threshold=8):
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'''Return the polynomials defining n-th homogeneous part of the canonical ideal.'''
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return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]
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as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials
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class as_symmetric_product_forms:
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def __init__(self, forms_and_coeffs):
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'''Elements of forms_and_coeffs are of the form [coeff, form_1, ..., form_n]'''
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self.n = len(forms_and_coeffs[0]) - 1
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forms_and_coeffs1 = []
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for atuple in forms_and_coeffs:
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if atuple[0] != 0 and atuple[1:] not in [a[1:] for a in forms_and_coeffs1]:
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forms_and_coeffs1 += [atuple]
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elif atuple[1:] in [a[1:] for a in forms_and_coeffs1]:
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i = [a[1:] for a in forms_and_coeffs1].index(atuple[1:])
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forms_and_coeffs1[i][0] += atuple[0]
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if len(forms_and_coeffs1) == 0:
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forms_and_coeffs1 = [[0] + forms_and_coeffs[0][1:]]
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self.tuples = forms_and_coeffs1
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self.curve = forms_and_coeffs1[0][1].curve
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def coordinates(self, basis = 0):
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AS = self.curve
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g = AS.genus()
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n = self.n
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F = AS.base_ring
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if basis == 0:
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basis = AS.holomorphic_differentials_basis()
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from itertools import product
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indices = [list(range(g)) for i in range(self.n)]
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result = {i : 0 for i in product(*indices)}
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for atuple in self.tuples:
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coors = [omega.coordinates(basis = basis) for omega in atuple[1:]]
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for i in product(*indices):
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aux_product = 1
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for j in range(n):
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aux_product *= coors[j][i[j]]
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result[i] += atuple[0]*aux_product
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return result
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def __repr__(self):
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result = ''
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for atuple in self.tuples:
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if atuple[0] !=0:
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result += str(atuple[0]) + ' * '
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for j in range(1, self.n+1):
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result += "(" + str(atuple[j]) + ")"
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if j != self.n:
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result + '⊗'
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result += ' + '
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return result
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def __add__(self, other):
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return as_symmetric_product_forms(self.tuples + other.tuples)
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def __rmul__(self, other):
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AS = self.curve
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F = AS.base_ring
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if isinstance(other, int) or other in ZZ or other in F:
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aux_tuples = []
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for atuple in self.tuples:
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atuple1 = [other * atuple[0]] + atuple[1:]
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aux_tuples += [atuple1]
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return as_symmetric_product_forms(aux_tuples)
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def multiply(self):
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'''The map H^0(Ω)^⊙n ---> H^0(Ω^⊗n)'''
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n = self.n
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AS = self.curve
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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result = as_polyform(0*AS.x, n)
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for atuple in self.tuples:
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aux_product = Rxyz(1)
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for fct in atuple[1:]:
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aux_product = aux_product * fct.form
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aux_product = as_function(AS, aux_product)
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aux_product = as_reduction(AS, aux_product)
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aux_product = as_function(AS, aux_product)
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result += as_polyform(atuple[0]*aux_product, n)
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return result
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def polynomial(self):
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'''Return the associated polynomial.'''
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AS = self.curve
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F = AS.base_ring
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g = AS.genus()
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M = self.coordinates()
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n = self.n
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Rg = PolynomialRing(F, 'X', g)
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X = Rg.gens()
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from itertools import product
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indices = [list(range(g)) for i in range(self.n)]
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result = Rg(0)
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for i in product(*indices):
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aux_product = Rg(1)
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for j in range(n):
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aux_product *= X[i[j]]
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result += M[i] * aux_product
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return result
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def group_action(self, elt):
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p = self.curve.base_ring.characteristic()
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n = self.curve.height
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aux_tuples = []
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for atuple in self.tuples:
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aux_tuple = [atuple[0]] + [a.group_action(elt) for a in atuple[1:]]
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aux_tuples += [aux_tuple]
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return as_symmetric_product_forms(aux_tuples)
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def non_decreasing(L):
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return all(x<=y for x, y in zip(L, L[1:]))
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def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
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'''Return the group action matrices for the n-th homogeneous part of the canonical ideal.'''
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K = as_canonical_ideal(AS, mult, threshold = threshold)
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n = AS.height
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F = AS.base_ring
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K_polynomials = [a.polynomial() for a in K]
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r = len(K)
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matrices = []
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for i in range(n):
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M = matrix(F, r, r)
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K_group_action_polynomials = [a.group_action([j == i for j in range(n)]).polynomial() for a in K]
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for i in range(r):
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M[i, :] = vector(linear_representation_polynomials(K_group_action_polynomials[i], K_polynomials))
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matrices += [M]
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return matrices
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as_cover.group_action_canonical_ideal = as_matrices_group_action_canonical_ideal |